Particle accelerators generally are grouped into different categories according to their fundamental concepts:
1) Those that use constant electrostatic fields such as Van de Graaff accelerators;
2) Those that make use of radiofrequency cavities in a straight line such as linear accelerators;
3) Those that use the electric fields induced by a time varying magnetic field to accelerate a particle such as the betatron; and
4) Circular accelerators that recirculate the beam of particles through a radiofrequency cavity to reach a desired energy such as a cyclotron, synchrotron, microtron, racetrack microtron or Rhodotron™.
Different names have been used to describe different combinations of the ideas represented by these groups and the concepts they represent as they have been perceived to be advantageous in different applications. Many are discussed in books about accelerator design such as M. S. Livingston and J. P. Blewett, “Particle Accelerators”, McGraw Hill Book Company, Inc., New York, 1962. They all apply the fundamental Maxwell equations and particle dynamics in magnetic and electric fields to accelerate particles and form accelerated beams.